eq. 05a: file eq05a.gif
eq. 05b: file eq05b.gif
the distance between the points being eq. 06 .
But the metre-rod is moving with the velocity v relative to K. It therefore
follows that the length of a rigid metre-rod moving in the direction of its length
with a velocity v is eq. 06 of a metre.
The rigid rod is thus shorter when in motion than when at rest, and the more
quickly it is moving, the shorter is the rod. For the velocity v=c we should have
eq. 06a ,
and for stiII greater velocities the square-root becomes imaginary. From this we
conclude that in the theory of relativity the velocity c plays the part of a limiting
velocity, which can neither be reached nor exceeded by any real body.
Of course this feature of the velocity c as a limiting velocity also clearly follows
from the equations of the Lorentz transformation, for these became meaningless
if we choose values of v greater than c.
If, on the contrary, we had considered a metre-rod at rest in the x-axis with
respect to K, then we should have found that the length of the rod as judged from
K1 would have been eq. 06 ;
this is quite in accordance with the principle of relativity which forms the basis
of our considerations.
A Priori it is quite clear that we must be able to learn something about the
physical behaviour of measuring-rods and clocks from the equations of
transformation, for the magnitudes z, y, x, t, are nothing more nor less than the
results of measurements obtainable by means of measuring-rods and clocks. If
we had based our considerations on the Galileian transformation we should not
have obtained a contraction of the rod as a consequence of its motion.
Let us now consider a seconds-clock which is permanently situated at the origin
(x1=0) of K1. t1=0 and t1=I are two successive ticks of this clock. The first and